Maximum Independent Sets in Subcubic Graphs: New Results

The maximum independent set problem is known to be NP-hard in the class of subcubic graphs, i.e. graphs of vertex degree at most 3. We present a polynomial-time solution in a subclass of subcubic graphs generalizing several previously known results

A note on anti-coordination and social interactions

This note confirms a conjecture of [Bramoull\'{e}, Anti-coordination and social interactions, Games and Economic Behavior, 58, 2007: 30-49]. The problem, which we name the maximum independent cut problem, is a restricted version of the MAX-CUT problem, requiring one side of the cut to be an independent set. We show that the maximum independent cut problem does not admit any polynomial time algorithm with approximation ratio better than $n^{1-\epsilon}$, where $n$ is the number of nodes, and $\epsilon$ arbitrarily small, unless P=NP. For the rather special case where each node has a degree of at most four, the problem is still MAXSNP-hard.Comment: 7 page

Hard and easy instances of L-Tromino tilings

In this work we study tilings of regions in the square lattice with L-shaped trominoes. Deciding the existence of a tiling with L-trominoes for an arbitrary region in general is NP-complete, nonetheless, we identify restrictions to the problem where it either remains NP-complete or has a polynomial time algorithm. First, we characterize the possibility of when an Aztec rectangle has an L-tromino tiling, and hence also an Aztec diamond; if an Aztec rectangle has an unknown number of defects or holes, however, the problem of deciding a tiling is NP-complete. Then, we study tilings of arbitrary regions where only 180∘ rotations of L-trominoes are available. For this particular case we show that deciding the existence of a tiling remains NP-complete; yet, if a region does not contain so-called “forbidden polyominoes” as subregions, then there exists a polynomial time algorithm for deciding a tiling

Independent Set Reconfiguration in Cographs

We study the following independent set reconfiguration problem, called TAR-Reachability: given two independent sets $I$ and $J$ of a graph $G$, both of size at least $k$, is it possible to transform $I$ into $J$ by adding and removing vertices one-by-one, while maintaining an independent set of size at least $k$ throughout? This problem is known to be PSPACE-hard in general. For the case that $G$ is a cograph (i.e. $P_4$-free graph) on $n$ vertices, we show that it can be solved in time $O(n^2)$, and that the length of a shortest reconfiguration sequence from $I$ to $J$ is bounded by $4n-2k$, if such a sequence exists. More generally, we show that if $X$ is a graph class for which (i) TAR-Reachability can be solved efficiently, (ii) maximum independent sets can be computed efficiently, and which satisfies a certain additional property, then the problem can be solved efficiently for any graph that can be obtained from a collection of graphs in $X$ using disjoint union and complete join operations. Chordal graphs are given as an example of such a class $X$

Maximum Independent Sets in Subcubic Graphs: New Results

International audienceWe consider the complexity of the classical Independent Set problem on classes of subcubic graphs characterized by a finite set of forbidden induced subgraphs. It is well-known that a necessary condition for Independent Set to be tractable in such a class (unless P=NP) is that the set of forbidden induced subgraphs includes a subdivided star S k,k,k , for some k. Here, S k,k,k is the graph obtained by taking three paths of length k and identifying one of their endpoints. It is an interesting open question whether this condition is also sufficient: is Independent Set tractable on all hereditary classes of subcu-bic graphs that exclude some S k,k,k ? A positive answer to this question would provide a complete classification of the complexity of Independent Set on all classes of subcubic graphs characterized by a finite set of forbidden induced subgraphs. The best currently known result of this type is tractability for S2,2,2-free graphs. In this paper we generalize this result by showing that the problem remains tractable on S 2,k,k-free graphs, for any fixed k. Along the way, we show that subcubic Independent Set is tractable for graphs excluding a type of graph we call an "apple with a long stem", generalizing known results for apple-free graphs

Parameterized Inapproximability of Independent Set in HH-Free Graphs

We study the Independent Set (IS) problem in $H$-free graphs, i.e., graphs excluding some fixed graph $H$ as an induced subgraph. We prove several inapproximability results both for polynomial-time and parameterized algorithms. Halld\'orsson [SODA 1995] showed that for every $\delta>0$ IS has a polynomial-time $(\frac{d-1}{2}+\delta)$-approximation in $K_{1,d}$-free graphs. We extend this result by showing that $K_{a,b}$-free graphs admit a polynomial-time $O(\alpha(G)^{1-1/a})$-approximation, where $\alpha(G)$ is the size of a maximum independent set in $G$. Furthermore, we complement the result of Halld\'orsson by showing that for some $\gamma=\Theta(d/\log d),$ there is no polynomial-time $\gamma$-approximation for these graphs, unless NP = ZPP. Bonnet et al. [IPEC 2018] showed that IS parameterized by the size $k$ of the independent set is W[1]-hard on graphs which do not contain (1) a cycle of constant length at least $4$, (2) the star $K_{1,4}$, and (3) any tree with two vertices of degree at least $3$ at constant distance. We strengthen this result by proving three inapproximability results under different complexity assumptions for almost the same class of graphs (we weaken condition (2) that $G$ does not contain $K_{1,5}$). First, under the ETH, there is no $f(k)\cdot n^{o(k/\log k)}$ algorithm for any computable function $f$. Then, under the deterministic Gap-ETH, there is a constant $\delta>0$ such that no $\delta$-approximation can be computed in $f(k) \cdot n^{O(1)}$ time. Also, under the stronger randomized Gap-ETH there is no such approximation algorithm with runtime $f(k)\cdot n^{o(k)}$. Finally, we consider the parameterization by the excluded graph $H$, and show that under the ETH, IS has no $n^{o(\alpha(H))}$ algorithm in $H$-free graphs and under Gap-ETH there is no $d/k^{o(1)}$-approximation for $K_{1,d}$-free graphs with runtime $f(d,k) n^{O(1)}$.Comment: Preliminary version of the paper in WG 2020 proceeding

On cycle transversals and their connected variants in the absence of a small linear forest.

A graph is H-free if it contains no induced subgraph isomorphic to H. We prove new complexity results for the two classical cycle transversal problems Feedback Vertex Set and Odd Cycle Transversal by showing that they can be solved in polynomial time for (sP1+P3) -free graphs for every integer s≥1 . We show the same result for the variants Connected Feedback Vertex Set and Connected Odd Cycle Transversal. For the latter two problems we also prove that they are polynomial-time solvable for cographs; this was known already for Feedback Vertex Set and Odd Cycle Transversal

Listing Maximal Independent Sets with Minimal Space and Bounded Delay

International audienceAn independent set is a set of nodes in a graph such that no two of them are adjacent. It is maximal if there is no node outside the independent set that may join it. Listing maximal independent sets in graphs can be applied, for example, to sample nodes belonging to different communities or clusters in network analysis and document clustering. The problem has a rich history as it is related to maximal cliques, dominance sets, vertex covers and 3-colorings in graphs. We are interested in reducing the delay, which is the worst-case time between any two consecutively output solutions, and the memory footprint, which is the additional working space behind the read-only input graph